Ponder This

September 2015

IBM decided to give an award for technical excellence to exactly N people.
Several teams submitted their nominations.
Many of them are single-person teams, but 8 are multi-person teams of more than 1 member.
It so happens that the size of the teams is such that awards can be given to any number of teams, from 1 to N, while keeping the total number of people exactly N.
Find the sizes of the 8 teams that enable the maximal N.

There is a better solution (N > 6) for 4 teams, but a possible answer to the same question with 4 multi-person teams and N=6 is: 2,3,3,6; since you can award a single team (6); 2 teams (3,3); 3 teams (1,2,3); 4 teams (1,1,1,3); 5 teams (1,1,1,1,2); and 6 teams (1,1,1,1,1,1).

Update (3/9):

You can do better than 56.

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